What is the derivative of x^{-8cos(x)} ?

The derivative of x^{-8cos(x)} is x^{-8cos(x)}(8sin(x)ln(x)-(8cos(x))/x)

What is the first derivative of x^{-8cos(x)} ?

The first derivative of x^{-8cos(x)} is x^{-8cos(x)}(8sin(x)ln(x)-(8cos(x))/x)

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derivative

x^{- 8 c o s (x)} (8 sin (x) ln (x) - \frac{8 cos ( x )}{x})

Solution steps

\frac{d}{d x} (x^{- 8 c o s (x)})

Apply exponent rule:

a^{b} = e^{b l n (a)}

x^{- 8 c o s (x)} = e^{(- 8 c o s (x)) l n (x)}

= \frac{d}{d x} (e^{(- 8 c o s (x)) l n (x)})

Apply the chain rule:

e^{(- 8 c o s (x)) l n (x)} \frac{d}{d x} ((- 8 cos (x)) ln (x))

= e^{(- 8 c o s (x)) l n (x)} \frac{d}{d x} ((- 8 cos (x)) ln (x))

\frac{d}{d x} ((- 8 cos (x)) ln (x)) = 8 sin (x) ln (x) - \frac{8 cos ( x )}{x}

= e^{(- 8 c o s (x)) l n (x)} (8 sin (x) ln (x) - \frac{8 cos ( x )}{x})

Simplify

e^{(- 8 c o s (x)) l n (x)} (8 sin (x) ln (x) - \frac{8 cos ( x )}{x}) : x^{- 8 c o s (x)} (8 sin (x) ln (x) - \frac{8 cos ( x )}{x})

= x^{- 8 c o s (x)} (8 sin (x) ln (x) - \frac{8 cos ( x )}{x})

What is the derivative of x^{-8cos(x)} ?
The derivative of x^{-8cos(x)} is x^{-8cos(x)}(8sin(x)ln(x)-(8cos(x))/x)
What is the first derivative of x^{-8cos(x)} ?
The first derivative of x^{-8cos(x)} is x^{-8cos(x)}(8sin(x)ln(x)-(8cos(x))/x)